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refactor(linear_algebra/charpoly): split file to reduce imports (#13778)
While working on representation theory I was annoyed to find that essentially all of field theory was being transitively imported (causing lots of unnecessary recompilation). This improves the situation slightly. Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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‎archive/100-theorems-list/83_friendship_graphs.lean

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@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Aaron Anderson, Jalex Stark, Kyle Miller
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-/
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import combinatorics.simple_graph.adj_matrix
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import linear_algebra.matrix.charpoly.coeff
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import linear_algebra.matrix.charpoly.finite_field
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import data.int.modeq
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import data.zmod.basic
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import tactic.interval_cases

‎src/linear_algebra/charpoly/basic.lean

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import linear_algebra.free_module.finite.basic
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import linear_algebra.matrix.charpoly.coeff
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import field_theory.minpoly
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/-!
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@@ -30,6 +31,7 @@ open_locale classical matrix polynomial
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noncomputable theory
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open module.free polynomial matrix
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namespace linear_map

‎src/linear_algebra/matrix/charpoly/coeff.lean

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Authors: Aaron Anderson, Jalex Stark
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-/
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import algebra.polynomial.big_operators
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import data.matrix.char_p
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import field_theory.finite.basic
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import group_theory.perm.cycles
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import data.polynomial.expand
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import linear_algebra.matrix.charpoly.basic
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import linear_algebra.matrix.trace
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import linear_algebra.matrix.to_lin
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import ring_theory.polynomial.basic
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import ring_theory.power_basis
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/-!
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# Characteristic polynomials
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not_false_iff] }
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end
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@[simp] lemma finite_field.matrix.charpoly_pow_card {K : Type*} [field K] [fintype K]
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(M : matrix n n K) : (M ^ (fintype.card K)).charpoly = M.charpoly :=
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begin
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casesI (is_empty_or_nonempty n).symm,
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{ cases char_p.exists K with p hp, letI := hp,
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rcases finite_field.card K p with ⟨⟨k, kpos⟩, ⟨hp, hk⟩⟩,
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haveI : fact p.prime := ⟨hp⟩,
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dsimp at hk, rw hk at *,
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apply (frobenius_inj K[X] p).iterate k,
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repeat { rw iterate_frobenius, rw ← hk },
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rw ← finite_field.expand_card,
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unfold charpoly, rw [alg_hom.map_det, ← coe_det_monoid_hom,
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← (det_monoid_hom : matrix n n K[X] →* K[X]).map_pow],
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apply congr_arg det,
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refine mat_poly_equiv.injective _,
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rw [alg_equiv.map_pow, mat_poly_equiv_charmatrix, hk, sub_pow_char_pow_of_commute, ← C_pow],
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{ exact (id (mat_poly_equiv_eq_X_pow_sub_C (p ^ k) M) : _) },
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{ exact (C M).commute_X } },
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{ -- TODO[gh-6025]: remove this `haveI` once `subsingleton_of_empty_right` is a global instance
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haveI : subsingleton (matrix n n K) := subsingleton_of_empty_right,
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exact congr_arg _ (subsingleton.elim _ _), },
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end
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@[simp] lemma zmod.charpoly_pow_card (M : matrix n n (zmod p)) :
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(M ^ p).charpoly = M.charpoly :=
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by { have h := finite_field.matrix.charpoly_pow_card M, rwa zmod.card at h, }
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lemma finite_field.trace_pow_card {K : Type*} [field K] [fintype K]
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(M : matrix n n K) : trace (M ^ (fintype.card K)) = trace M ^ (fintype.card K) :=
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begin
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casesI is_empty_or_nonempty n,
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{ simp [zero_pow fintype.card_pos, matrix.trace], },
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rw [matrix.trace_eq_neg_charpoly_coeff, matrix.trace_eq_neg_charpoly_coeff,
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finite_field.matrix.charpoly_pow_card, finite_field.pow_card]
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end
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lemma zmod.trace_pow_card {p : ℕ} [fact p.prime] (M : matrix n n (zmod p)) :
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trace (M ^ p) = (trace M)^p :=
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by { have h := finite_field.trace_pow_card M, rwa zmod.card at h, }
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namespace matrix
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theorem is_integral : is_integral R M := ⟨M.charpoly, ⟨charpoly_monic M, aeval_self_charpoly M⟩⟩
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theorem minpoly_dvd_charpoly {K : Type*} [field K] (M : matrix n n K) :
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(minpoly K M) ∣ M.charpoly :=
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minpoly.dvd _ _ (aeval_self_charpoly M)
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/-- Any matrix polynomial `p` is equivalent under evaluation to `p %ₘ M.charpoly`; that is, `p`
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is equivalent to a polynomial with degree less than the dimension of the matrix. -/
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lemma aeval_eq_aeval_mod_charpoly (M : matrix n n R) (p : R[X]) :
@@ -243,29 +190,3 @@ lemma pow_eq_aeval_mod_charpoly (M : matrix n n R) (k : ℕ) : M^k = aeval M (X^
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by rw [←aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
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end matrix
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section power_basis
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open algebra
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/-- The characteristic polynomial of the map `λ x, a * x` is the minimal polynomial of `a`.
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In combination with `det_eq_sign_charpoly_coeff` or `trace_eq_neg_charpoly_coeff`
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and a bit of rewriting, this will allow us to conclude the
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field norm resp. trace of `x` is the product resp. sum of `x`'s conjugates.
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-/
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lemma charpoly_left_mul_matrix {K S : Type*} [field K] [comm_ring S] [algebra K S]
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(h : power_basis K S) :
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(left_mul_matrix h.basis h.gen).charpoly = minpoly K h.gen :=
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begin
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apply minpoly.unique,
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{ apply matrix.charpoly_monic },
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{ apply (injective_iff_map_eq_zero (left_mul_matrix _)).mp (left_mul_matrix_injective h.basis),
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rw [← polynomial.aeval_alg_hom_apply, aeval_self_charpoly] },
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{ intros q q_monic root_q,
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rw [matrix.charpoly_degree_eq_dim, fintype.card_fin, degree_eq_nat_degree q_monic.ne_zero],
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apply with_bot.some_le_some.mpr,
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exact h.dim_le_nat_degree_of_root q_monic.ne_zero root_q }
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end
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end power_basis

‎src/linear_algebra/matrix/charpoly/finite_field.lean

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/-
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Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Aaron Anderson, Jalex Stark
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-/
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import linear_algebra.matrix.charpoly.coeff
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import field_theory.finite.basic
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import data.matrix.char_p
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/-!
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# Results on characteristic polynomials and traces over finite fields.
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-/
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noncomputable theory
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open polynomial matrix
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open_locale polynomial
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variables {n : Type*} [decidable_eq n] [fintype n]
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@[simp] lemma finite_field.matrix.charpoly_pow_card {K : Type*} [field K] [fintype K]
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(M : matrix n n K) : (M ^ (fintype.card K)).charpoly = M.charpoly :=
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begin
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casesI (is_empty_or_nonempty n).symm,
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{ cases char_p.exists K with p hp, letI := hp,
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rcases finite_field.card K p with ⟨⟨k, kpos⟩, ⟨hp, hk⟩⟩,
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haveI : fact p.prime := ⟨hp⟩,
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dsimp at hk, rw hk at *,
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apply (frobenius_inj K[X] p).iterate k,
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repeat { rw iterate_frobenius, rw ← hk },
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rw ← finite_field.expand_card,
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unfold charpoly, rw [alg_hom.map_det, ← coe_det_monoid_hom,
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← (det_monoid_hom : matrix n n K[X] →* K[X]).map_pow],
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apply congr_arg det,
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refine mat_poly_equiv.injective _,
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rw [alg_equiv.map_pow, mat_poly_equiv_charmatrix, hk, sub_pow_char_pow_of_commute, ← C_pow],
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{ exact (id (mat_poly_equiv_eq_X_pow_sub_C (p ^ k) M) : _) },
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{ exact (C M).commute_X } },
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{ -- TODO[gh-6025]: remove this `haveI` once `subsingleton_of_empty_right` is a global instance
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haveI : subsingleton (matrix n n K) := subsingleton_of_empty_right,
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exact congr_arg _ (subsingleton.elim _ _), },
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end
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@[simp] lemma zmod.charpoly_pow_card {p : ℕ} [fact p.prime] (M : matrix n n (zmod p)) :
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(M ^ p).charpoly = M.charpoly :=
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by { have h := finite_field.matrix.charpoly_pow_card M, rwa zmod.card at h, }
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lemma finite_field.trace_pow_card {K : Type*} [field K] [fintype K]
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(M : matrix n n K) : trace (M ^ (fintype.card K)) = trace M ^ (fintype.card K) :=
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begin
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casesI is_empty_or_nonempty n,
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{ simp [zero_pow fintype.card_pos, matrix.trace], },
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rw [matrix.trace_eq_neg_charpoly_coeff, matrix.trace_eq_neg_charpoly_coeff,
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finite_field.matrix.charpoly_pow_card, finite_field.pow_card]
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end
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lemma zmod.trace_pow_card {p : ℕ} [fact p.prime] (M : matrix n n (zmod p)) :
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trace (M ^ p) = (trace M)^p :=
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by { have h := finite_field.trace_pow_card M, rwa zmod.card at h, }

‎src/linear_algebra/matrix/charpoly/minpoly.lean

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/-
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Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Aaron Anderson, Jalex Stark
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-/
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import linear_algebra.matrix.charpoly.coeff
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import ring_theory.power_basis
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/-!
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# The minimal polynomial divides the characteristic polynomial of a matrix.
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-/
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noncomputable theory
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universes u v
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open polynomial matrix
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variables {R : Type u} [comm_ring R]
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variables {n : Type v} [decidable_eq n] [fintype n]
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open finset
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variable {M : matrix n n R}
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namespace matrix
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theorem is_integral : is_integral R M := ⟨M.charpoly, ⟨charpoly_monic M, aeval_self_charpoly M⟩⟩
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theorem minpoly_dvd_charpoly {K : Type*} [field K] (M : matrix n n K) :
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(minpoly K M) ∣ M.charpoly :=
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minpoly.dvd _ _ (aeval_self_charpoly M)
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end matrix
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section power_basis
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open algebra
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/-- The characteristic polynomial of the map `λ x, a * x` is the minimal polynomial of `a`.
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In combination with `det_eq_sign_charpoly_coeff` or `trace_eq_neg_charpoly_coeff`
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and a bit of rewriting, this will allow us to conclude the
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field norm resp. trace of `x` is the product resp. sum of `x`'s conjugates.
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-/
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lemma charpoly_left_mul_matrix {K S : Type*} [field K] [comm_ring S] [algebra K S]
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(h : power_basis K S) :
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(left_mul_matrix h.basis h.gen).charpoly = minpoly K h.gen :=
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begin
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apply minpoly.unique,
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{ apply matrix.charpoly_monic },
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{ apply (injective_iff_map_eq_zero (left_mul_matrix _)).mp (left_mul_matrix_injective h.basis),
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rw [← polynomial.aeval_alg_hom_apply, aeval_self_charpoly] },
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{ intros q q_monic root_q,
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rw [matrix.charpoly_degree_eq_dim, fintype.card_fin, degree_eq_nat_degree q_monic.ne_zero],
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apply with_bot.some_le_some.mpr,
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exact h.dim_le_nat_degree_of_root q_monic.ne_zero root_q }
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end
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end power_basis

‎src/ring_theory/norm.lean

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import field_theory.primitive_element
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import linear_algebra.determinant
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import linear_algebra.matrix.charpoly.coeff
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import linear_algebra.matrix.charpoly.minpoly
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import linear_algebra.matrix.to_linear_equiv
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import field_theory.is_alg_closed.algebraic_closure
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‎src/ring_theory/trace.lean

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-/
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import linear_algebra.matrix.bilinear_form
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import linear_algebra.matrix.charpoly.coeff
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import linear_algebra.matrix.charpoly.minpoly
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import linear_algebra.determinant
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import linear_algebra.vandermonde
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import linear_algebra.trace

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